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The exams relate to an airman certificate (private pilot, commercial pilot, instructor pilot, etc.). You have passed the private pilot exam and you have been awarded a certificate. If you wish to remain as a private pilot you would not need to take another exam. You can add other ratings (single-engine land, multiengine land) to your certificate by passing an appropriate checkride with an examiner. This would require instruction and an endorsement from a CFI that you are prepared for the checkride. Keep in mind that the examiner will check your knowledge prior to the flight. This might include anything that was on the private written test. I expect at a minimum you would need to be able to demonstrate weight and balance calculation, obtaining weather, METAR and TAF, be able to explain all the symbols on your sectional map, and know your airspace. Good luck.
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